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Results tagged with sequences-and-series
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user 109816
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
3
votes
Accepted
Sum of digits of all 4-digit numbers divisible by 7
There are 9 1-digit decimal numbers. Partition them on their residue modulo 7 and you get:
$$(0,\{7\}), (1,\{1,8\}), (2,\{2,9\}), (3,\{3\}), (4,\{4\}), (5,\{5\}), (6,\{6\})$$
We can extetnd this to …
- 4,236
0
votes
Sum of infinite sequence?
Since $\ln (1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4}\dots$, your series is equivalent to $\ln 1.5 = \ln 3 - \ln 2 \approx 0.4054651081081645$.
It's an alternating harmonic series …
- 4,236
0
votes
Algorithm to detect how many times a sequence alternates
The obvious algorithm is $O(n)$ and the problem requires at least $O(n)$ work since every element has to be examined.
- 4,236
0
votes
Convergence of series with ^(n+1)
I think this is the piece you're missing:
$$\begin{align}
\lim_{n \rightarrow \infty} \frac{(n+1)^n}{n^n} & = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n\\
\ & = e
\end{align}$$
- 4,236
0
votes
Can anyone prove the identity $\sum_{m=-\infty}^\infty (z+\pi m)^{-2} = (\sin z)^ {-2} $
I don't have a solution, but I have a simplification.
I'd start with:
$$\frac{d}{dx} -\cot(z) = \sin^{-2}z$$
$$\int \frac{1}{(z + \pi m)^2}dz = -\frac{1}{z + \pi m}$$
So then you would just need to …
- 4,236
4
votes
How to prove $\sum_{n=0}^\infty \frac1{n!}=e\ $?
First look at the derivative of $2^x$.
$$\begin{align}
\frac{d}{dx} 2^x & = \lim_{\delta \rightarrow 0} \frac{2^{x+\delta} - 2^x}{\delta}
\\ & = 2^x \lim_{\delta \rightarrow 0} \frac {2^\delta - 1}{\ …
- 4,236
11
votes
The series of reciprocals of the integers that do not contain 9 in their decimal representation
Let $c_n$ be the sum of all reciprocals of $n$-digit positive numbers that contain no '9' digit. There are $8 \cdot 9^{n-1}$ $n$-digit numbers that contain no '9' digit. The largest term is the firs …
- 4,236
2
votes
Accepted
Clarification: Proof of the quotient rule for sequences
If, for eny $\epsilon > 0 \in \mathbb{R}$, there exists an $n \in \mathbb{Z}$ such that $\forall i > n, |a_i - c| < \epsilon $, then $c$ is defined as the limit of the sequence of $a_i$.
Assuming the …
- 4,236
2
votes
If $C_0,C_1,C_2,\cdots C_n$ denotes the binomial coefficients in the expansion of $(1+x)^n$ ...
$\sum_{r=0}^n \sum_{s=0}^n C_s = (n+1)2^n$. Then you have $2(n+1)2^n = (n+1)2^{n+1}$
- 4,236
3
votes
Accepted
Number of Fibonacci series that contain a certain integer
Like Edward said, your problem breaks down to finding solutions to $F_na+F_{n+1}b = k$ for a given $k$.
If you're willing to consider negative $a$ and $b$, then there are an infinite number of solu …
- 4,236
0
votes
What is this sequence of polynomials?
Let $q_n(a,b) = \sum_{k_1=a}^b \sum_{k_2 = k_1}^b \dots \sum_{k_n=k_{n-1}}^b k_1 k_2 \dots k_n$.
$p_n(x) = q_n(1,x)$
$q_1(a,b) = \dfrac{b(b+1)}{2} - \dfrac{a(a-1)}{2} $
$q_n(a,a) = a^n$
$q_{n+1}(a …
- 4,236
1
vote
Does there exist a fraction whose repeating decimal has a period of some highest number or c...
$\frac{1}{10^p - 1}$ will have period $p$ for any positive integer $p$.
- 4,236